About Halfway There

Equivalent Fractions
Equivalent Fractions

I’ve been interested in creating fractions projects for kids exactly as long as I’ve been working with children in schools (decades). This year, after enjoying,messing around with a hexagon/golden ratio project I wondered if I could modify the idea of using scaled hexagons to help fourth graders make better sense of fractions. My first attempt at this didn’t work out so well.

Making 1, or 100%
Making 1, or 100% out of two halves

I gave students hexagons that were scaled to 1, one-half, a third, a fourth, a fifth, a sixth, an eighth, a tenth, and twelfths. The task was to pair and arrange them so they would span the length of a whole, aka 100% across.

1/3 + 1/6 + 1/6 + 1/3 = 1
1/3 + 1/6 + 1/6 + 1/3 = 1

The project went okay, but it just didn’t snap for me.

I’ve been thinking about how to improve this project. Today I had a chance to work with a small group of kids. I tried a new approach that worked so much better. What was especially great was that it included making a simple book. Yay!

Fractions for a Book
Fractions for a Book

I started kids off with a hexagon that was labeled 1/2. I explained about how the lengths we would be looking at would be the horizontal or vertical length of the hexagon (I didn’t use these words, rather gestured what I meant). Then we layered the hexagon with equivalencies. Here you can see two 1/12ths equals 1/6, three 1/6ths equals 1/2, and 1/6th and two 1/12ths equals 1/3.

Equivalent Fractions
Equivalent Fractions

Nice, right? Snap!

The books we made were just two sheets of paper folded in half, bound with yarn using a modified pamphlet stitch. 

Equivalent Fractions Book
Equivalent Fractions Book

What’s great about using hexagons for this project is that you can still see the labels of the lower layers as the equivalencies are built up. The adults in the room had a bit of trouble with accepting that the hexagons were scaled (similar) versions of each other, but the kids had no problem with it. This reinforces my notion that children have a better intuitive understanding of scale than do adults.

This is the way I explain the scaling to adults: We all know what half a candy bar looks like. That’s one way of thinking of one-half. But when we say a child is half the size of the parent, we don’t envision the child to be half a parent, like they were half a candy bar. Instead, we envision them smaller than the parent in their height as well as width.  This explanation seems to work.

The bullseye view of fractions
The bullseye view of fractions

After doing a bunch of equivalencies, this child decided to nest her fractions.

Okay then. Here, what’s obvious is the hierarchy of the hexagons that are scaled by fractions. Nice!

This project can use a bit more refinement, but this is as far as I’m going with it right now.

I’m including PDF of the hexagons. The labeling includes the colors of the paper I use for printing.  Yeah, it’s lots of files. Welcome to my life.


hexagon halves blue

hexagon sixths halve grape

hexagon 3rds 12ths chartreusegreen

hexagon 4ths orange

hexagon 5ths 8ths pink

hexagon 6ths halves grape

zhexagon 12ths full page chartreuse

2 pieces to make 12 inch hexagon

12 inch hexagon

Notes about hexagons




2 thoughts on “About Halfway There

    1. When introduce this project I do discuss that we are looking at distance rather than area. The kids seem completely comfortable with this but with the adults, not so much. It’s great to be able to have this conversation with adults, which is really about dilation versus scale. I try to be responsive to the discomfort this can cause, but I think the benefits of the scaled hexagon model outweighs its cons. BTW, I don’t really use the word dilation when I talk about this, as I hardly ever see it used. Paul Lockhart’s book, Measurement, talks about scale being two dilations(page 41). So If I had a hexagon that was a dilation by a factor of .5, then we’d end up with a hexagon with .5 of the original area, but it wouldn’t work well (at least I don’t think so) with this project.


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